Proposal for revision of ISO 4377: Flat-V weirs

(1)

Proposal for reV1Sl0n of ISO 4377 Flat-V weirs

J.P.Th. Kalkwijk Wang Lianxiang

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LaboratoryDepartment oof Civil Engineeringf Fluid Mechanics

(2)

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Proposal for revision of ISO 4377

Flat-'V' weirs

J.P.Th. Kalkwijk

Wang Lianxiang

Report no. 19 - 83

Laboratory of Fluid Mechanics

Department of Civil Engineering

Delft University o

f

Technology

(3)

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Elucidation to the Netherlands' proposal for revision of ISO 4377

,

Liquid flow measurements in open channels

Flat-V weirs

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In ISO/TC 113/SC 2/308 it was suggested that ISO 4377 could be

significantly simplified as far as the calculation of discharge

was involved.

The proposal for revision is based upon this

suggestion.

The revision ~s fully bàsed upon the material given in ISO 4377

.

In that respect no changes have been made.

The result of the revision is~ that

i

the calculation of discharge is simple and straightforward

ii

the number of tables (sometimes very extensive) has reduced

from 13 to only 5.

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(4)

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7.

7. 1

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1 I

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· I

7.2 I

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7.3 I

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..

-1-Discharge equation Equation

The equation of discharge ~s based on the use of gauged head:

5/

1/

.

IJ

5/

Q

=

(t)

2

(t)

2 CDCvCSCdr m g 2h' 2 (1)

,where

Q

is the discharge

CD is the discharge coefficient

C is the velocity of approach factor

v

Cs is the shape factor

Cdr is tbe drowned flow reduction factor

m is tbe crest cross-slope

g ~s the acceleration due to gravity

h is the upstream gauged head relative to the lowest crest

elevation.

Discharge coefficient

The discharge coefficient 1S g~ven by

CD

=

CDm<hè/h)5/3

=

CDm(1 - k

m

/h)5/2 ₍₂₎

where

CDm is the modular coefficient of discharge,

~~ is the effective gaugeg head, which is equal to (h-km),

k is the he ad correction factor~

m

Both CUm and ~ depend on the cross-slope of the weir. They are

given in table 3.

In the case of field measurement, km ~s negligible because it ~s

less than 1 mmo

Velocity of approach factor

Since the discharge equation ~s d~rived from a formula in which

the total head is used, a Vèlocity of approach factor is introduced

to correct the gauged head. That is

C

=

(R/h) 5

l :

v (3)

where

R is the upstream total head with respect to the lowest crest

elevation.

(5)

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,

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'

1 I

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7.4

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7.5 I

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-2-(4), where

PI

1S the difference between the lowest crest elevation and the

mean bed level in the approach channel.

b is the crest width

r

(5)

Equation (4) can be solved iteratively; the values of C; are

v

obtained in terms of YI and given in table 4.

In the case of

Y

I <

0,08

(Cv-I)« I, a good approximation for

CV 1S given below

C v

1, 2SY I

=

I +

-=--=-=-

_1-2,SYI

(6)

The relative error due to this approximation 1S less than 0,7 7.

Shape factor

A

shape factor is introduced into the discharge equation for

flat-V weir because the geometry of flow changes when the

discharge exceeds the V-full condition. Thus:

if

h

_{e_}<

h'

(8)

if h, > h'

e

where h' = h/2m is the difference between the lowest and highest

crest elevations.

Drowned flow reduction factor

When the weir gets drowned, i.e., h >0,4He, the discharge decreases.

pe

(6)

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.~.

-3

-1

1

Cdr := C .

=

dr if h /R < 0,4 pe e-1,078j-0,909-Ch

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H

.

)3/2

J

0,I8_3ifh /H > 0,4 _ pe e pe e (9) where

·

1

h

pe is the effective separation pocket head relative to the lowest crest

elevation, h :=h - k •

pe p m

i$ the gauged separation pocket head relative to the lowest crest

elevation.

is the effective upstream total head relative to the lowest

2

crest elevation,

H

:=h + v /2g - k •

e m

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1

h p H e

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Raving solved equation (9), the values of Cdr are obtained and

given in Table 5, in terms of two parameters.

Y2(:= cDCsmh2/bCPl+h).

h [t«, and

pe e

1

In case of hpe/he < 0,9 Cdr can be calculated by us~ng two-step

procedure with a good approximation (relative error <

1 %

).

This

method is as follows:

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a) To estimate the value of CdrO according to the value of h /h.,

pe e CdrO 0,I if h /h < 0,55 pe e CdrO := 0,9 if 0,55 < h /h. < 0,70 - pe e _(J₀₎ Cdro := 0,8 if 0,70 < h /h < 0,85 - pe e Cdro :=0,75 if 0,85 < h /he < 0,90 - pe

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b) To calculate the following quantities,

CDCSmh2 .. Y 2 :=b(PI+h)

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Y1 :=0,8 _c_I_ro y2 2 CO,4 I + 0,5 2 (1 I) := _YI_C v v He :=CO_v,4 h_e I,078~, 909

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O•183 Cdr := - (hpe/Re)3/2

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'When h Ihe> 0,9, more steps of iteration are needed and table 5

pe

-is recorrrrnended.

(7)

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7.6

7.6.1 I

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7.6.2 I

:1

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-4-Limits of app1ication

The practical 10wer limit of upstream head is re1ated to the magnitude of the inf1uence of f1uid properties and boundary roughness. For a we11-maintained weir with a smooth crest section, the minimum head recommended is 0,03 m. If the crest is of smooth concrete or a material of simi1ar texture, a 10wer limit of 0,06-m is suggested.

There is a1so a 1imiting va1ue for the ratio h'/PI of

2,5

and t.here are limitations on HdP2 as shown in t'able 3. These are governed by the scope of experi~ta1 verification and vary with cross-s10pe. P2 is the e1evation of the 10west crest e1evation relative to the downstream bed level.

(8)

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-5-I

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8.

8. 1 Computation of discharge Computation steps

Equation(l) is used for computing discharges from the known

quantities •. Pltm,h' ,b,h and h. The calculation proceeds as

p

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fola) lows:Calculate h "= h - k and CD using equation (2) and the

e m

appropriate va lues of CD and k from table 3.

m m

Calculate Cs using equation

(8).

~alculate Cdr. If it is in modular flow comditions, Cdr=)·

"If it is in drowned flow conditions, calculate h =h-k

pe p m

of k from table 3; calculate

m Y2 = CDCS

mh

2/b(Pj+hY; look up using the

..

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b) c)

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the ratio appropriate value h /h; calculate pe e

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the appropriate value of Cdr from table 5, or calculate Cdr

according to equations (10) and (11)~

d) Calculate Yl uS1ng equation (5); Determine the value of C by

v

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e) Determineusing equationthe value of(6) or table 4.

Q

by substituting the known and the

calculated values into equation (1).

Examples are given in clause 10.

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8.2

Accuracy

8.2.1

The overall accuracy of measurement wil1 depend on:

a) The accuracy of construction and finish of the weir;

b) the accuracy of the head measurements;

c) the accuracy of other measured dimensions;

d) the accuracy of the coefficient values;

e) the accuracy of the form of the discharge equation.

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The method by which estimates of these constituent uncertainties

may be combined to give the overall uncertainty 1n computed discharges

is given in clause 9.

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_8.4.2

_Th_e _unc_e_rtainties _{(95% con}_f_idence _{limits) on modular discharge}

coefficients are given in table 3. These reflect the random and

systematic errors which occur 1n calibration experiments and also

the real but marginal changes 1n coefficient values which occur

changing discharge.

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(9)

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-6-I

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9. Errors in flow measurement

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9.1

General

9.1.1

The uncertainty of any flow measurement can be estimated if the

uncertainties from various sourced are combined. These contributions

to the total uncertainty may be assessed and will indicat.e whether

the rate of flow can be measured with sufficient accuracy for the

purpose in hand. This clause prbvides sufficient information for

estimating the uncertainties of measurements of discharge.

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9.1.2

The error in a result is the difference hetween.the true rate of

flow and that calculated using the discharge equations quoted in

this International Standard. Thus the error is, by definition,

unknown.but the uncertainty of the measurement may be estimated.

The term uncertainty denotes the deviation from the true rate of

flow with which the measured rate of flow is expected to lie some

nineteen times out of twenty (the 957. confidence limits).

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~.-".'-""- - ~.

9.2

Sources of error

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9.2.)

The sourees of error in the discharge measurement can be identified

byeons idering the form of equa tion (1) i.e ,

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(1)

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The constant

(t)5/2 <t)1/2

~s not subject to error and error ~n g

may be ignored. Hence the sources of error which need to he considered

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are:

a) The discharge coefficient CD which essentially has the same value

of uncertainty as C

Dm'

given in tahle 3.

The velocity of approach factor C. The following approximate

v

expression may be used to determine the uncertainty in C •

v

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b)

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Xc

=

0,5 hiP)

(%)

v

(12)

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c) The shape factor cS.uncertainty in this value.When heWhen

2

h',he >

Cs

h', the= 1 and there ~s novalue of

Cs

depends on h' and h , see equation (8). Both these quantities

e

will he of reasonahle magnitude and errors ~n heads will not

normally be significant at this stage. Thus the uncertainty

~n

Cs

~s negligible, i.e. X

Cs = o.

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(10)

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-1-I

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d) The.crest cross-slope, m. Nlimerical values will dep end on

the 'accuracY'of construction and subsequent measurement of the

structure.

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e) The upstream gauged head, h. The uncertainty 1n h depends on

uncertainties in head measurement, zeroing of the gauge and

uncertainties associated with the number of readings. Thus

lOOVeh2+eh 2+(2Sh-)2' o (%) h

=

+ (13)

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whei1e

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f)

eh is the uncertainty in the measurement of upstream head.

Any uncertainty which does not change randomly during a series

of measurements should be included here for example backlash

and friction;

eh is the uncertainty in the determination of the gauge zero;

'0

2Sh

is the uncertainty in the mean of n readings of upstream

head, see 9.3. It is associated with the random fluctuation 1n

a series of measurements.

The uncertainties eh and e~ dep end on an assessment of probable

uncertainties by the user.

The separation pocket head, h The uncertainty in hp depends

p

on uncertainties in head measurement, zeroing of the gauge and

uncertainties assosiated with the number of readings.

Thus

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IOO~h + eh + (2S

h

)2

~

;

= .:_

--_.!,..P---,h-p__:.o----fl---{%) (14)

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where

eh 1S the uncertainty in the measurement of the separation

p

pocket head. The sources of errors are systematic, for example

backlash and friction;

eh is the uncertainty 1n the determination of the gauge zero;

o

2Shp is the uncertainty in the mean of n readings of the

separation pocket head, see 9.3. It is associated with the

random fluctuatl.ons in a series of measurements.

The uncertainties eh and eh depend on an estimate of probable

p 0

errors by the user. If only one reading of head is made, then

the random uncertainties 2S- or 2S- must be estimated, see 9.5.4.

h hp

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(11)

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-8-I

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g) The drowned flow reduction factor, Cdr' There are three factors

which influence the uncertainty 1n Cdr: the uncertainty iri the

laboratory determination of the Cd versus h /He·relationship;

r pe

·The uncertainty in the measurement of the upstream head, h;

and the uncertainty in the measurement of the separation. pocket

head, hp •

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A suitable expression for the combined uncertainty

is :

"

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(15)

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9.3

Random and systematic errorsErrors can be classified as random or systematic, the former

affecting the reproducibility of measurement and the lat ter affecting

its true accuracy.

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9.3.1

The standard deviation of a set of n measurements of a variabie

R may be estimated from the equation

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~

n -

~lh

S =

L

(y - y)

YIn - 1 (16)

1

where y is the observed mean.

is then given by

The standard deviation of the mean

I

_S

s-

= J_

y

Ii..

(J 7)

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and the uncertainty of the mean 1S

2S-

y (for 95% probability) if

the number of readings, n, is large.

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·

9.3.2

A measurement can also be subject to systematic errors and the

mean of a large number of measured values would still differ from

the true value of the quantity being measured. An error in a

gauge zero, for example, will produce a systematic error. As

repetition of the measurements does not eliminate systematic

errors, the actual value could only be determined by an independent

measurement known to be more accurate.

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(12)

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-9-I

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9.4

Errors in quantities given in this International Standard

9.4.1

All the errors in this category are systematic. The values of the discharge coefficients, etc., quoted in this International Standard are .based on an appraisal of experiemnts, carefully carried out with sufficient repetition of readings.

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9.4.2

However, when measurements are made on other similar installation, systematic discrepancies between coefficients of discharge may occur due to variation in the surface finish of the device, its installation, the appr9ach flow conditions, etc •.

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9.4.3

The probable uncertainties in the coefficients quoted in previous clauses of this International Standard are based on a consideration of the deviation of experimental data from the given working

equations and a comparison of the equations themselves.

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9.5

Errors in quantities measured by the user.

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9.5.J

Both random and systematic errors will occur in measurements 1n this category.

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_9.5.2

_{Since neither the methods} _o_f _measurement _{nor the way} _{in which} _they

are to be made are specified, no numerical values for uncertainties in this catègory can be given.

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9.5.3

The uncertainties in the gauged head should be determined from an assessment of the separate soureed of uncertainty, for example the gauge sensitivity, the zero uncertainty, temperature effects, the backlash in the indicating mechanism, the residual random uncertainty in the mean of a series of measurements, etc.

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9.5.4

The above component uncertainties should be calculated as percentage standard deviations at the

95%

confidence limits but when the value of the component uricertainty 1S determined from only a single

measurement, the uncertainty 1S said to be rectangular distributed and may be taken, for the purposes of this International Standard, the limits (plus or minus) within the true value is known to lie with certainty (i.e. half the estimated maximum deviation).

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(13)

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-10-I

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9.6

Combination of uncertainties to gl.ve the overall uncertainty in discharge

9.6.1

The uncertainty in discharge is given by the expression

+

v'

X 2 1 .X

Q

=

_- _C + X2 + X2 + X2 + 6.25

x{

(l8) Cv _Cdr m D

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where XQ is the uncertainty.in the ccimputed discharge (per cent).

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9.6.2 It should be noted that the uncertainty in discharge is not single valued for a given device, but varies with flow. It may therefore be necessary to consider the uncertainty at several discharges covering the required range of measurement.

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(14)

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-11-I

10. Examples

10.1 Modular flow at low discharge (he

<

hl)

lö.i;] À

flat-V weir has a crest cross-slope of 1 : 20,3.

The crest

width and approach channel width are both 36 m an9 the mean upstream

bed level is 0,82 m below the lowest

· crest deviation.

Calculate the discharge when the observed upstream gauged head is

0,62 m.

Ten successive readings of this head prod~ce

· a standard

deviation of the mean of 0,5

mm

and the estimated uncertainty in

the gauge zero ~s 1

mmo

The basic measurements with their estimated

uncertainties are given below:

I

m

=

20,30

(.:!:_

0,2

i. )

b

₌

36,00 m

(+

0,005 m)

p~

=

0,82 m

(.:!:_

0,00] m)

h

₌

0,621 m

(.:!:_

0,003 m)

hl

₌

_{0,887 m}

₍₊

0,001 m)

eh

₌

.:!:. 0,001 m

0

2S-

=

t

0,001 m

h

:""

.

r>••

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The appropriate coefficient and head correction values are obtained

from table 3, as follows:

CDm

=

1,22

(.:!:_

3,2

i.)

k

=

0,0005 m

(.:!:_

0,0002 m)

m

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10.1.2 Calculation of discharge

(See8.1)

a)

_he

=

h - k

=

0,6205 m

m

5/

CD

CDm(he/h)

2 =

0,6188

Cs

=

1,0

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b)

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c) d)

I

=

0,0056

C

=

+

1,25YL

=

1,0071

v l"'72;)Y

1

5/ 1/ 1/2 5/2 = (%) 2 (~)

2 CDCvCSCdr m

g

h

e)

_Q

3 =

9,63 m /s

3

(15)

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1 -12-1

10.1.3 Uncertainty in calculated discharge

(See 9.2)

]0.2

Drowned flow at high discharge

10.2.1 A flat-V weir has a crest cross-slope of 1 : 10.1.

The crest

width and the approach channel width are both 25 m, and the mean

upstream bed level is 0,56 m below the lowest crest elevation.

Calculate the discharge when the upstream and crest tappings record

heads of 2,614 mand

2,211 m respectively.

Five successive readings

of the upstream head produc~d a standard deviation of the mean

of 1,5

mm

and the estimated uncertainty in the gauge zero is 2

mmo

Five successive readings of the separation pocket head produce a

standard deviation of the mean of 2,1 mm and the estimated uncertainty

in the gauge zero is also 2

mmo

The basic measurements with their

estimated uncertainties are g~ven below:

1

1 I

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1 I

1

1 I

·

1 I

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a)

From table 3 Xc

=

+ 3,2 %.

D

From equation (]2)

Xc

= ~

0,5 h/P1

(%)

v

= ~

0,38 %

.: b) c) d) e)

For modular flow X

Cdr

= +

0,2 %

=

o.

From data X

.

m

From equation (13)

100~h2+eh~+(2Sh)2

~ =

+ h

(%)

~ ~ 0,53 %

f)

By using equation (18), we have

X

Q

=

+

I

X 2 + X 2 + X 2

.

+ X2 + 6 25 ~2

-

CD

Cv

Cdr

m

'

-n

=

.:!:.

3,42 %

Thus the uncertainty in discharge (95% confidence limits)

is

.:!:.

3,42 %.

m

'

₌

10,1

_(.:!:.

0,2 %)

b

₌

25,00 m

(~ 0,004 m)

PI

=

0,56 m

(+ 0,002 m)

h

₌

2,614 m

(_!

0,003 m)

hp

₌

2,211 m

(~ 0,003 m)

(16)

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-13-I

h'

eh

o

2sii

2S-hp

=

1,238

m

C+

0,001

m)

=

.!.

0,002

m = +

0,003 m

=

+

0,0042

m

,I

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The appropriate coefficient and head correction values are obtained

from table 3,

as follows:

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C

Dm

=

1,22

k '=

0,0008

m m

c.:!:.

2,3

%) C.!.

0,0002

m)

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10.2.2 Calculation of discharge

(See

8.1)

a)

h

· =

h -

k

=

2,6132 m,

say 2,613 m

e m

CD

=

C

Dm

Che./h)512

=

0,6194

t", "

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b) Cs

=

I - C1 - ~ ) 512

he

=

0,7991

,g:.) h'_"';;'h - k =

2,2102

m

say

2,210

m,

pe

_

p m

h

lhe

=

0,8458

pe

Y2

=

C

DCS

mh

2jbCP

i

+h)

=

0,4304

from table 5, Cdr

=

0,7880 Cinterpolated)

I

d),

I

=

0,0736

C

= v +

1,25Y

I

1-2,5Y

1

=

1,1127

I

e)

I

=

121,35

m3js

Thus th

e

co

m

puted dischar

g

e

is

121,35

m2js.

(17)

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-14-I

10.2.3 Uncertainty in calculated discharge

a)

From table 3

Xc

= ~

2,3 %

D

b) Xc

= ~

0,5 h/Pl

(%)

v

=

2,33 %

.

c)

From drowned flow

I

=

+- 0,18

%

100/eh

2 +ehz+(

2 S- )z

=

+ p 0 hp hp

= ~

0,25 %

(%)

From equation (15)

Xc = ~ 5 (1 -

Cdr)

ft

+ ~ +

~p

dr

(%)

=~1,11%

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d)

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e)

X

=

+ /

X

2

+ X

2 +

X 2

+ X

2+

6,25

x.

2 Q - CD Cv Cdr . m

"n

= ~

vi

2,3

2 +

2,3

2

+1,11

2

+0,2

2

+6,25

X

0,18

2

= ~ 3,49 %

I

Thus the uncertainty in the calculated discharge (95% confidence

~

limits)is~ 3,49

x.

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(18)

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Ab

I

BC

I

CDC

Dm

Cdr

I

Cs C v

I

eh eh 0

I

hg

I

HH max h

I

h'p k m

I

KI,KZ m

I

n P

I

Q

s-I

vh

I

Xc XD Cv

Xc

dr

I

~

I

XY1Q

I

YZ

I

Symbols and dimensions

Area of cross-section of flow

Crest width

Width of approach chennel

.Coefficient

Dischar.ge coefficient

Modular coefficient of discharge Drowned flow reduction factor Shape factor

Velocity of approach factor

Uncertainty in the head measurement

Uncertainty in the gauge zero

Acceleration due to gravity

Gauged head above lowest crest deviation Total head above lowest crest deviation

non-dimensional non-dimensional non-dimensional non-dimensional non-dimensional non-dimensional L L

L/tZ

L L Maximum upstream total head above lowest crest elevation L

Separation pocket head L

Difference between lowest and highest crest elevations L

Head correction factor L

non-dimensional Constants

Crest cross-slope (1 vertical /m.horizontal)

Number of measurements in a set

Difference between mean bed level and lowest crest

elevation L

Discharge L3/t

non-dimensional

Standard deviation of mean of several head readings L

Mean velocity at cross-section L/t

Percentage uncertainty 1n discharge coefficient non-dimensional

Percentage uncertainty 1n velocity factor non-dimensional

Percentage uncertainty in drowned flow reduction

factor

Percentage uncertainty 1n head measurement

Percentage uncertainty in discharge measurement

Parameter 1n calculation of velocity factor

Parameter 1n calculation of drowned flow reduction

factor

Coriolis energy coefficient

non-dimensional non-dimensional non-dimensional non-dimensional non-dimensional non-dimensional

(19)

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1 I

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·

1 I

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1 I

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Subscript 2

denotes upstream values

denotes downstream values

denotes "effective" and i:mplies that corrections for fluid

effects have been made to the quantity.

(20)

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... J•••

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Table 3. Summary of recommended coefficients., limitations and uncertainties

FLAT-V weir

1.

R/h' <

1,0

Modular coefficient, CDm

x

Read correction factor,

Km

Uncertainty in coefficient,Xc Dm Modular Limit Other limitations:

h'/Pl

h'

lP

2 Upstream tapping Crest cross-slope

1 : 10

1 : 20

1:40

or less _.. ~

1,23

1,22

.

-

1,21

0,0004

m

0,0005

m

0,0008

m +

3,0 %

~+

3,2 %

.!.

2,9 %

-

-6S

to

7S %

6S

to

7S %

65

to

75 %

-""...•

-

~...__.,

~2,5

<

2,5

~2,5

-10

h'

10

h'

10

h' 2. R/h' >

1,0

Modular coefficient, C

x

Dm Read correction factor,

Km

Uncertainty in coefficient, Xc

Dm Modular limi t Other limitations:

h'/P(

h'

lp

2 Upstream tapping

1

· ,24

1,23

1,22

0,0004

m

0,0005

m

0,0008

m

.!.

2,5 %

.!.

2,8 %

.!.

2,3

%

65

to

75

%

65

to

75 %

65

to

75

% <

2,5

<

2,5

<

2,5

-

-<

8,2

<

8,2

<

4,2

-

-10 h'

10 h'

;::

Computations under non~modular conditions should be based on CDm

=

1,25

,

1,24

and

1,22

respectively.

(21)

I

Y1 0,000 0,002 0,004 0,006 0,008 0,00 1,000 1,002 1,005 1,008 1,010 0,01 _1,

°

₁₃ 1,016 1,018 1,021 1,.024 0,02 1,027 1,030 1,032 1,035 1,038 0,03 1,041 1,044 . 1,047 1,050 1,054 0,04 1,057 1,060 1,063 1,067 1,070 0,05 1,074 1,077 1,080 1,084 1,088 0,06 1,092 1,096 1,100 1,104 1, 108 0,07 1,112 1,116 1, 120 1,124 1,129 0,08 1,133 1,138 1,143 1,148 1,153 0,09 1,158 1,163 1,168 1,174 1,180 0,10 1. ]85 1,191 ] , 197 '1,203 _{] ,2]}

°

0,11 ] ,216 1,223 1,230 1,237 1,244 0,12 1,252 1,260 1,268 1,277 1,286 0,13 1,295 1,305 1,316 1,326 1,338 0,14 1,350 1,363 1,377 1,392 1,408 0,15 1,426 1,446 1,468 1,492 1,523

I

.

I

· I

I

Table 4 Velocity of approach coefficient in terms of Y1

I

· I

I

(22)

I. I

Table 5 Drowned flowreduction factor,in termsof h·_p-e/h_e andY2

I

~

0.050 0.100 0.150 0.200 0.220 0,240 0.260 0.280 0.300 0.320 0.340 0.360 0.380 0.400 0.420 0.440 e 0.41 0.995 0.996 0.996 0.997 0.997 0.998 0.998 0.998 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 0.42 0.993 0.993 0.993 0.994 0.994 0.995 0.995 0.996 0.996 0.997 0.998 0.998 0.999 1.000 1.000 1.000 0.43 0.990 0.990 0.991 0.991 0.992 0.992 0.993 0.993 0.994 0.994 0.995 0.996 0.996 0.997 0.998 1.000 0.44 0.987 0.987 0.988 0.989 0.989 0.989 0.990 0.990 0.991 0.992 0.992 0.993 0.994 0.995 0.996 0.997 0.45 0.984 0.984 0.985 0.986 0.986 0.987 0.987 0.988 0.988 0.989 0.990 0.990 0.991 0.992 0.993 0.994 0.46 0.981 0.981 0.982 0.983 0.983 0.984 0.984 0.985 0.985 0.986 0.987 0.988 0.988 0.989 0.991 0.992 0.47 0.977 0.978 0.979 0.980 0.980 0.981 0.•981 0.982 0.982 0.983 0.984 0.985 0.986 0.987 0.988 ·0.989 0.48 0.975 0.975 0.976 0.977 0.977 0.978 0.978 0.979 0.979 0.980 0.981 0.982 0.983 0.984 0.985 0.986 0.49 0.972 0.9.72 0.973 0.973 0.974 0.974 0.975 0.976 0.976 0.977 0.978 0.979 0.980 0.981 0.982 0.984 0.50 0.968 0.969 0.969 0.970 0.971 0.971 0.972 0.972 0.973 0.974 0.975 0.976 0.977 0.978 0.979 0.981 0.51 0.965 0.965 0.966 0.967 0.967 0.968 0.969 0.969 0.970 0.971 0.972 0.973 0.974 0.975 0.976 0.978 0.52 0.961 0.962 0.962 0.963 0.964 0.965 0.965 0.966 0.967 0.967 0.968 0.969 0.971 0.972 0.973 0.975 0.53 0.958 0.958 0.959 0.960 0.960 0.961 0.962 0.962 0.963 0.964 0.965 0.966 0.967 0.969 0.970 0.972 0.54 0.954 0.954 0.955 0.956 0.957 0.957 0.958 0.959 0.960 0.961 0.962 0.963 0.964 0.965 0.967 0.968 0.55 0.950 0.951 0.951 0.953 0.953 0.954 0.954 0.955 0.956 0.957 0.958 0.959 0.961 0.962 0.963 0.965 0.56 0.946 0.947 0.948 0.949 0.949 0.950 0.95T 0.952 0.952 0.953 0.954 0.9% 0.957 0.958 0.960 0.962 0.57 0.942 0.943 0.944 0.945 0.945 0.946 0.947 0.948 0.949 0.950 0.951 0.952 0.953 0.955 0.956 0.958 0.58 0.938 0.939 0.940 0.941 0.941 0.942 0.943 0.944 0.945 0.946 0.947 0.948 0.950 0.951 0.953 0.955 0.59 0.934 0.934 0.935 0.937 0.937 0.938 0,939 0.940 0.941 0.942 0.943 0.944 0.946 0.947 0.949 0.951 0.60 0.929 0.930 9.931 0.932 0.933 0.934 0.935 0.936 0.937 0.938 0.939 0.940 0.942 0.943 0.945 0.947 0.61 0.925 0.926 0.927 0.928 0.929 0.929 0.930 0.931 0.932 0.934 0.935 0.936 0.938 0.939 0.941 0.943 0.62 0.920 0.921 0.922 0.923 0.924 0.925 0.926 0.927 0.928 0.929 0.931 0.932 0.934 0.935 0.937 0.939 0.63 0.916 0.916 0.917 0.919 0.920 0.920 0.921 0.922 0.923 0.925 0.926 0.928 0.929 0.931 0.933 0.935 0.64 0.911 0.911 0.912 0.914 0.915 0.916 0.917 0.918 0.919 0.920 0.922 0.923 0.925 0.927 0.929 0.931 0.65 0.905 0.906 0.907 0.909 0.910 0.911 0.912 0.913 0.914 0.915 à.917 0.918 0.920 0.922 0.924 0.927 0.66 0.900 0.901 0.902 0.904 0.905 0.906 0.907 0.908 0.909 0.910 0.912 0.914 0.915 0.917 0.920 0.922 0.67 0.895 0.895 0.897 0.898 0.899 0.900 0.901 0.903 0.904 0.905 0.907 0.909 0.911 0.913 0.915 0.•917 0.68 0.889 0.890 0.891 0.893 0.894 0.895 0.896 0.897 0.899 0.900 0.902 0.903 0.905 0.908 0.910 0.912 0.69 0.883 0.884 0.885 0.887 0.888 0.889 0.890 0.892 0.893 0.895 0.896 0.898 0.900 0.902 0.905 0.907 0.70 0.877 0.878 0.879 0.881 0.882 0.883 0.885 0.886 0.887 0.889 0.891 0.893 0.895 0.897 0.899 0.902 0.71 0.871 0.872 0.873 0.875 0.876 0.877 0.878 0.880 0.881 0.883 0.885 0.887 0.889 0.891 0.894 0.897 0.72 0.864 0.865 0.867 0.869 0.870 0.871 0.872 0.874 0.875 0.877 0.879 0.881 0.883 0.886 0.888 0.891 0.73 0.857 0.858 0.860 0.862 0.863 0.864 0.866 0.867 0.869 0.871 0.873 0.875 0.877 0.880.0.882 0.885 0.74' 0.850 0.851 0.853 0.855 0.856 0~857 0.859 0.860 0.862 0.864 0.866 0.868 0.871 0.873 0.876 0.879 0.75 0.843 0.844 0.845 0.848 0.849 0.850 0.852 0.853 0.855 0.857 0.859 0.861 0.864 0.867 0.870 0.873 0.76 0.835 0.836 0.837 0.840 0.841 0.843 0.844 0.846 0.848 0.850 0.852 0.854 0.857 0.860 0.863 0.866 0.77 0.826 0.827 0.829 0.832 0.833 0.835 0.836 0.838 0.840 0.842 0.844 0.847 0.849 0.852 0.856 0.859 0.78 0.818 0.819 0.821 0.823 0.825 0.826 0.828 0.830 0.832 0.834 0.836 0.839 0.842 0.845 0.848 0.852 0.79 0.808 0.810 0.812 0.814 0.816 0.817 0.819 0.821 0.823 0.825 0.828 0.831 0.834 0.837 0.840 0.844 0.80 0.799 0.800 0.802 0.805 0.806 0.808 0.810 0.812 0.814 0.816 0.819 0.822 0.825 0.828 0.832 0.836 0.81 0.788 0.789 0.792 0.795 0.796 0.798 0.800 0.802 0.804 0.807 0.810 0.813 0.816 0.819 0.823 0.827 0.82 0.777 0.778 0.781 0.784 0.786 0.788 0.790 0.792 0.794 0.797 0.800 0.803 0.806 0.810 0.814 0.818 0.83 0.765 0.766 0.769 0.772 0.774 0.776 0.778 0.781 0.783 0.786 0.789 0.792 0.796 0.799 0.804 0.808 0.84 0.752 0.754 0.756 0.760 0.762 .0.764 0.766 0.768 0.771 0.774 0.777 0.781 0.784 0.788 0.793 0.797 0.85 0.738 0.739 0.742 0.746 0.748 0.750 0.753 0.755 0.758 0.761 0.765 0.768 0.772 0.776 0.781 0.786 0.86 0.722 0.724 0.727 0.731 0.733 0.736 0.738 0.741 0.744 0.747 0.751 0.755 0.759 0.763 0.768 0.773 0.87 0.705 0.707 0.710 0.715 0.717 0.719 0.722 0.725 0.728 0.732 0.736 0.740 0.744 0.749 0.754 0.760 0.88 0.685 0.687 0.691 0.696 0.698 0.701 0.704 0.707 0.711 0.714 0.719 0.723 0.728 0.733 0.738 0.745 0.89 0.662 0.665 0.669 0.674 0.677 0.680 0.683 0.687 0.690 0.695 0.699 0.704 0.709 0.715 0.721 0.727 0.90 0.635 0.638 0.642 0.649 0.6.52 0.655 0.659 0.662 0.667 0.671 0.676 0.682 0.688 0.694 0.700 0.708 0.91 0.602 0.605 0.610 0.617 0.620 0.624 0.628 0.633 0.638 0.643 0.649 0.655 0.662 0.669 0.676 0.684 0.92 0.556 0.560 0.566 0.575 0.579 0.584 0.589 0.595 0.600 0.607 0.614 0.621 0.629 0.637 0.646 0.655 0.93 0.483 0.488 0.497 0.510 0.516 0.522 0.529 0.537. 0.545 0.553 0.563 0.572 0.582 0.593 0.604 0.615